Solve 12^3x-5=40

$$12 ^ { 3 x - 5 } = 40$$

Solve for x

$$x = \frac{\log_{12} {(9953280)}}{3} \approx 2.161504766$$

Steps Using Definition of Logarithm

Use the rules of exponents and logarithms to solve the equation.

$$12^{3x-5}=40$$

Take the logarithm of both sides of the equation.

$$\log(12^{3x-5})=\log(40)$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$\left(3x-5\right)\log(12)=\log(40)$$

Divide both sides by \log(12).

$$3x-5=\frac{\log(40)}{\log(12)}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$3x-5=\log_{12}\left(40\right)$$

Add 5 to both sides of the equation.

$$3x=\log_{12}\left(40\right)-\left(-5\right)$$

Divide both sides by 3.

$$x=\frac{\log_{12}\left(40\right)+5}{3}$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{3\ln(12)}+\frac{\log_{12}\left(9953280\right)}{3}$$

$$n_{1}\in \mathrm{Z}$$